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NASA NTRS1983

The stability of coronal loops - Finite-length and pressure-profile limits

Results are described from a quickly converging, necessary-and-sufficient, MHD-stability test for coronal-loop models. The primary stabilizing influence arises from magnetic line tying at the photosphere, and this end conditions requires a series expansion of possible loop excitations. The stability boundary is shown to quickly approach a limit as the number of terms increases, providing a critical length for the loop in proportion to its transverse magnetic scale. Several models of force-free-field profiles are tested and the stability behavior of a localized current channel, embedded in an external current-free region, is shown to be superior to that of other, broader, current profiles. Pressure-gradient effects, leading to increased or decreased stability, are shown to be amplified by line tying. Long loops must either conduct low net current, or exhibit an axial-field reversal coexisting with a low-pressure core. The limits on stability depend on the magnetic aspect ratio, the plasma-to-magnetic pressure ratio, and the field orientation at the loop edge. Applications of these results to the structure of coronal loops are described.

Einaudi, G.↗

NASA NTRS1982

Toward a better understanding of helicopter stability derivatives

An amended six degree of freedom helicopter stability and control derivative model was developed in which body acceleration and control rate derivatives were included in the Taylor series expansion. These additional derivatives were derived from consideration of the effects of the higher order rotor flapping dynamics, which are known to be inadequately represented in the conventional six degree of freedom, quasistatic stability derivative model. The amended model was a substantial improvement over the conventional model, effectively doubling the usable bandwidth and providing a more accurate representation of the short period and cross axis characteristics. Further investigations assessed the applicability of the two stability derivative model structures for flight test parameter identification. Parameters were identified using simulation data generated from a higher order base line model having sixth order rotor tip path plane dynamics. Three lower order models were identified: one using the conventional stability derivative model structure, a second using the amended six degree of freedom model structure, and a third model having eight degrees of freedom that included a simplified rotor tip path plane tilt representation. Previously announced in STAR as N82-32376

Hansen, R. S.↗

NASA NTRS1984

Toward a better understanding of helicopter stability derivatives

An amended six degree of freedom helicopter stability and control derivative model was developed in which body acceleration and control rate derivatives were included in the Taylor series expansion. These additional derivatives were derived from consideration of the effects of the higher order rotor flapping dynamics, which are known to be inadequately represented in the conventional six degree of freedom, quasistatic stability derivative model. The amended model was a substantial improvement over the conventional model, effectively doubling the unstable bandwidth and providing a more accurate representation of the short period and cross axis characteristics. Further investigations assessed the applicability of the two stability derivative model structures for flight test parameter identification. Parameters were identified using simulation data generated from a higher order base line model having sixth order rotor tip path plane dynamics. Three lower order models were identified: one using the conventional stability derivative model structure, a second using the amended six degree of freedom model structure, and a third model having eight degrees of freedom that included a simplified rotor tip path plane tilt representation. Previously announced in STAR as N82-32376

Hansen, R. S.↗

NASA NTRS1984

A method for computing the kernel of the downwash integral equation for arbitrary complex frequencies

For the design of active controls to stabilize flight vehicles, which requires the use of unsteady aerodynamics that are valid for arbitrary complex frequencies, algorithms are derived for evaluating the nonelementary part of the kernel of the integral equation that relates unsteady pressure to downwash. This part of the kernel is separated into an infinite limit integral that is evaluated using Bessel and Struve functions and into a finite limit integral that is expanded in series and integrated termwise in closed form. The developed series expansions gave reliable answers for all complex reduced frequencies and executed faster than exponential approximations for many pressure stations.

Desmarais, R. N.↗

NASA NTRS1984

Three-dimensional Navier-Stokes calculations of multiple interacting vortex rings

Results from a finite-difference Navier-Stokes code for three-dimensional, unsteady, vortical flows in unbounded domains are presented and analyzed in this paper. The vortical flows presented are representative of vortex rings and other closed vortical tubes or structures in fluid mechanics. Such structures are important elements in fluid flows such as jets, atmospheric turbulence, and the far-field wakes of aircraft, and studies of their interaction may aid in an understanding of complex fluid flows. The paper demonstrates that computational methods can be used as a viable alternative or supplement to experimental techniques for studying the physics of vortex flows. The separate visualization of vortex stretching, convection, and diffusion is presented in this paper for a single elliptical vortex ring.The calculations employ a truncated series expansion technique to simulate the unbounded nature of the fluid flow with a finite computational domain, which is a more accurate technique than the conventional freestream boundary specification. The numerical divergence of the three-dimensional vorticity field is considered as a useful estimate of truncation error, and the use of a kinetic energy decay law as a calculation check is demonstrated. Results from the Navier-Stokes code are presented for the unsteady motion of two and four vortex rings along parallel axes, and the results agree qualitatively with experimental flow visualization.

Chamberlain, J. P.↗

NASA NTRS1984

Aberrations of varied line-space grazing incidence gratings in converging light beams

Analyses of the imaging properties of several designs for varied-line space gratings in converging beams of light in grazing-incidence spectrometers are presented. An explicit model is defined for the case of a plane-reflection grating intercepting light that converges and is reflected to a stigmatic point associated with the zero-order image of the grating. Smooth spatial variation of the grating constant then permits aberration correction. The aberrations are expressed as polynomials in the grating lens coordinates using power series expansions. Application of the model is illustrated in terms of aberrations experienced with the short wavelength spectrometer on the EUVE satellite. Attention is given to straight and parallel in-plane grooves, curved groove in-plane designs and off-plane grooves. Aberrations due to dispersions and misalignment are also considered.

Hettrick, M. C.↗

NASA NTRS1984

Shear deformation plate continua of large double-layered space structures

The energy equivalence to construct equivalent continua for the actual lattice structure is described. An energy equivalent continuum is defined as that which has the same amount of strain and kinetic energies stored in it as the original lattice structure when both are subjected to the same loading conditions. The equivalent continuum is characterized by its strain and kinetic energies from which the constitutive relations and the equations of motion can be derived. A simple method to model large rigid jointed lattices as continuous media with couple stresses is presented. The transition from the discrete system to the continuous medium is achieved by expanding the displacements and the rotations of the nodal points in a Taylor series about a suitable chosen origin. Basic kinematic assumptions are introduced to insure that the assumptions used in deriving the governing equations of the modeled continuum are satisfied. The number of terms retained in the Taylor series expansion will depend upon the properties to be evaluated. This implies that the kind of continuum needed to model from the discrete lattice, before the actual properties are derived was to predetermined.

Hefzy, M. S.↗

NASA NTRS1984

Kinematics of Nearby Gas and Stars

The kinematic properties of gas and stars in the solar neighborhood are described in terms of the line of sight component of a three dimensional first order Taylor series expansion of the local velocity field. The types of object analyzed are: (1) 21 cm absorbing clouds; (2) intercloud medium; (3) main sequence B-stars closer than 200 pc; (4) B-stars of luminosity class ranging from I to IV; (5) main sequence A stars; (6) K-giant stars. The least squares fitting procedure used to derive the 10 coefficients describes each of the six velocity fields was essentially the same for comparison. Marked departures from circular motion are found in most cases, but the only systematic trend is a correlation between delta u/delta x and stellar spectral type, where the gas behaves like a medium younger than the early type stars. It is indicated that the standard plane-parallel model provides a good description for the intercloud medium, but is inadequate for the absorbing clouds. A velocity ellipsoid description of the residuals is presented for each type of object. The influence of the Gould belt on local kinematics is discussed.

Goulet, T.↗

NASA NTRS1983

Difference equation technique for determining the evolution of radially symmetric ultrasonic fields

A simplified difference equation technique has been developed to calculate the three-dimensional evolution of radially symmetric, complex ultrasonic fields. The parabolic approximation to the Helmholtz equation is obtained by assuming that localized changes in the magnitude of the field with respect to distance along the axis of propagation are small compared to those with respect to radial displacement. The Helmholtz equation in cylindrical coordinates may then be approximated, after separating variables, using truncated series expansions. The resulting difference equation determines complex field values at discrete locations in a plane using corresponding adjacent locations in the preceding plane. Stability criteria which determine acceptable values for various parameters are discussed. The technique has been applied to the propagation of ultrasonic fields generated by two-dimensional focusing and non-focusing, uniform and Gaussian profile, piezoelectric transducers in both profiled and non-profiled media.

Dockery, G. D.↗

NASA NTRS1986

STEP and STEPSPL: Computer programs for aerodynamic model structure determination and parameter estimation

The successful parametric modeling of the aerodynamics for an airplane operating at high angles of attack or sideslip is performed in two phases. First the aerodynamic model structure must be determined and second the associated aerodynamic parameters (stability and control derivatives) must be estimated for that model. The purpose of this paper is to document two versions of a stepwise regression computer program which were developed for the determination of airplane aerodynamic model structure and to provide two examples of their use on computer generated data. References are provided for the application of the programs to real flight data. The two computer programs that are the subject of this report, STEP and STEPSPL, are written in FORTRAN IV (ANSI l966) compatible with a CDC FTN4 compiler. Both programs are adaptations of a standard forward stepwise regression algorithm. The purpose of the adaptation is to facilitate the selection of a adequate mathematical model of the aerodynamic force and moment coefficients of an airplane from flight test data. The major difference between STEP and STEPSPL is in the basis for the model. The basis for the model in STEP is the standard polynomial Taylor's series expansion of the aerodynamic function about some steady-state trim condition. Program STEPSPL utilizes a set of spline basis functions.

Batterson, J. G.↗

NASA NTRS1986

Supercomputer implementation of finite element algorithms for high speed compressible flows

Prediction of compressible flow phenomena using the finite element method is of recent origin and considerable interest. Two shock capturing finite element formulations for high speed compressible flows are described. A Taylor-Galerkin formulation uses a Taylor series expansion in time coupled with a Galerkin weighted residual statement. The Taylor-Galerkin algorithms use explicit artificial dissipation, and the performance of three dissipation models are compared. A Petrov-Galerkin algorithm has as its basis the concepts of streamline upwinding. Vectorization strategies are developed to implement the finite element formulations on the NASA Langley VPS-32. The vectorization scheme results in finite element programs that use vectors of length of the order of the number of nodes or elements. The use of the vectorization procedure speeds up processing rates by over two orders of magnitude. The Taylor-Galerkin and Petrov-Galerkin algorithms are evaluated for 2D inviscid flows on criteria such as solution accuracy, shock resolution, computational speed and storage requirements. The convergence rates for both algorithms are enhanced by local time-stepping schemes. Extension of the vectorization procedure for predicting 2D viscous and 3D inviscid flows are demonstrated. Conclusions are drawn regarding the applicability of the finite element procedures for realistic problems that require hundreds of thousands of nodes.

Thornton, E. A.↗

NASA NTRS1986

A finite element approach for solution of the 3D Euler equations

Prediction of thermal deformations and stresses has prime importance in the design of the next generation of high speed flight vehicles. Aerothermal load computations for complex three-dimensional shapes necessitate development of procedures to solve the full Navier-Stokes equations. This paper details the development of a three-dimensional inviscid flow approach which can be extended for three-dimensional viscous flows. A finite element formulation, based on a Taylor series expansion in time, is employed to solve the compressible Euler equations. Model generation and results display are done using a commercially available program, PATRAN, and vectorizing strategies are incorporated to ensure computational efficiency. Sample problems are presented to demonstrate the validity of the approach for analyzing high speed compressible flows.

Thornton, E. A.↗

NASA NTRS1986

Analytic computation of energy derivatives - Relationships among partial derivatives of a variationally determined function

Formulas are presented relating Taylor series expansion coefficients of three functions of several variables, the energy of the trial wave function (W), the energy computed using the optimized variational wave function (E), and the response function (lambda), under certain conditions. Partial derivatives of lambda are obtained through solution of a recursive system of linear equations, and solution through order n yields derivatives of E through order 2n + 1, extending Puley's application of Wigner's 2n + 1 rule to partial derivatives in couple perturbation theory. An examination of numerical accuracy shows that the usual two-term second derivative formula is less stable than an alternative four-term formula, and that previous claims that energy derivatives are stationary properties of the wave function are fallacious. The results have application to quantum theoretical methods for the computation of derivative properties such as infrared frequencies and intensities.

King, H. F.↗

NASA NTRS1986

N-point correlations for biased galaxy formation

A general series expansion of arbitrary accuracy is given for the correlation function of the 'biased' regions of galaxy formation in terms of the mass autocorrelations. It is shown how to calculate the N-point correlations to arbitrary accuracy, and the validity of the various approximations is discussed. The asymptotic values of the N-point correlation functions at their highest point at zero spatial separation are given, and the scaling of these values is discussed.

Jensen, L. G.↗

NASA NTRS1987

Separating Peaks in X-Ray Spectra

Deconvolution algorithm assists in analysis of x-ray spectra from scanning electron microscopes, electron microprobe analyzers, x-ray fluorescence spectrometers, and like. New algorithm automatically deconvolves x-ray spectrum, identifies locations of spectral peaks, and selects chemical elements most likely producing peaks. Technique based on similarities between zero- and second-order terms of Taylor-series expansions of Gaussian distribution and of damped sinusoid. Principal advantage of algorithm: no requirement to adjust weighting factors or other parameters when analyzing general x-ray spectra.

Nicolas, David↗

NASA NTRS1986

The nonlinear development of Gortler vortices in growing boundary layers

The development of Gortler vortices in boundary layers over curved walls in the nonlinear regime is investigated. The growth of the boundary layer makes a parallel flow analysis impossible except in the high wavenumber regime so in general the instability equations must be integrated numerically. Here the spanwise dependence of the basic flow is described using Fourier series expansion whilst the normal and streamwise variations are taken into account using finite differences. The calculations suggest that a given disturbance imposed at some position along the wall will eventually reach a local equilibrium state essentially independent of the initial conditions. In fact, the equilibrium state reached is qualitatively similar to the large amplitude high wave-number solution described asymptotically by Hall (1982). In general, it is found that the nonlinear interactions are dominated by a mean field type of interaction between the mean flow and the fundamental. Thus, even though higher harmonics of the fundamental are necessarily generated, most of the disturbance energy is confined to the mean flow correction and the fundamental. A major result of the calculations is finding that the downstream velocity field develops a strongly inflection character as the flow moves downstream. The latter result suggests that the major effect of Gortler vortices on boundary layers of practical importance might be to make them highly receptive to rapidly growing Rayleigh modes of instability.

Hall, Philip↗

NASA NTRS1986

Approximate Cartesian state transition matrix

Based on a Taylor series expansion, an easily-computed approximation to the Cartesian state transition matrix is presented for a general velocity-independent force field. Suitable for the short time intervals encountered in onboard navigation applications of the extended Kalman filter, it provides approximately five decimal digits of accuracy for earth orbiting spacecraft with update intervals of one minute, and better accuracy for shorter intervals.

Markley, F. L.↗

NASA NTRS1986

Analytic evaluation of two-center molecular integrals

By using the Fourier-transform technique, the explicit expressions for the one-electron - two-center overlap integrals of Slater-type atomic orbitals up to 3d are derived. The final expressions are analytic, simple, and independent of local coordinates. Furthermore, they do not contain the nonclosed-form of exponential integrals which were presented in expressions given in earlier work. It is shown that the two-electron - two-center Coulomb integrals, as well as the hybrid integrals, can simply be expressed in terms of these integrals. The numerical instability arising from the situation in which the exponents of the two orbitals are almost equal is discussed, and a solution for this problem based on a Taylor-series expansion of the integral is suggested.

Tai, H.↗